An experiment utilizing a TlF molecular beam is being developed by the CeNTREX collaboration to search for hadronic interactions that violate both time-reversal ($\mathcal{T}$) and parity ($\mathcal{P}$) invariance. Here, we propose to use the same beam to look for a $\mathcal{T}$-invariance conserving but $\mathcal{P}$-nonconserving (PNC) effect induced by the anapole moment of the Tl nucleus, via a vector coupling of the two nuclear spins in TlF. To measure the nuclear anapole moment, the dc electric and magnetic fields in CeNTREX are replaced by rf fields resonant with a nuclear spin-flip transition. We adapt the relativistic coupled-cluster method in combination with relativistic density functional theory for the calculation of the molecular PNC spin-spin vector coupling constant that links the experimental signal with the anapole moment. The value of the $\mathcal{P}$-conserving spin-spin coupling constant calculated within the same approach is found to be in good agreement with available experimental data.